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Convergence analysis of Wirtinger Flow for Poisson phase retrieval

Bing Gao, Ran Gu, Shigui Ma

TL;DR

This work provides a rigorous convergence analysis for Wirtinger Flow applied to Poisson phase retrieval, establishing linear convergence in the noiseless case and stability under bounded noise for a Poisson observation model with background terms. It extends WF with an Incremental Wirtinger Flow variant that processes one measurement per iteration and provably converges to the true signal with high probability, enhancing scalability. The paper develops smoothness and curvature conditions to underpin the convergence results and demonstrates that the Poisson-aligned WF outperforms Gaussian-model WF under Poisson noise, while the reverse holds under Gaussian noise. Collectively, the results offer a theoretically solid, computationally efficient approach for robust phase retrieval in photon-counting imaging contexts and related applications.

Abstract

This paper presents a rigorous theoretical convergence analysis of the Wirtinger Flow (WF) algorithm for Poisson phase retrieval, a fundamental problem in imaging applications. Unlike prior analyses that rely on truncation or additional adjustments to handle outliers, our framework avoids eliminating measurements or introducing extra computational steps, thereby reducing overall complexity. We prove that WF achieves linear convergence to the true signal under noiseless conditions and remains robust and stable in the presence of bounded noise for Poisson phase retrieval. Additionally, we propose an incremental variant of WF, which significantly improves computational efficiency and guarantees convergence to the true signal with high probability under suitable conditions.

Convergence analysis of Wirtinger Flow for Poisson phase retrieval

TL;DR

This work provides a rigorous convergence analysis for Wirtinger Flow applied to Poisson phase retrieval, establishing linear convergence in the noiseless case and stability under bounded noise for a Poisson observation model with background terms. It extends WF with an Incremental Wirtinger Flow variant that processes one measurement per iteration and provably converges to the true signal with high probability, enhancing scalability. The paper develops smoothness and curvature conditions to underpin the convergence results and demonstrates that the Poisson-aligned WF outperforms Gaussian-model WF under Poisson noise, while the reverse holds under Gaussian noise. Collectively, the results offer a theoretically solid, computationally efficient approach for robust phase retrieval in photon-counting imaging contexts and related applications.

Abstract

This paper presents a rigorous theoretical convergence analysis of the Wirtinger Flow (WF) algorithm for Poisson phase retrieval, a fundamental problem in imaging applications. Unlike prior analyses that rely on truncation or additional adjustments to handle outliers, our framework avoids eliminating measurements or introducing extra computational steps, thereby reducing overall complexity. We prove that WF achieves linear convergence to the true signal under noiseless conditions and remains robust and stable in the presence of bounded noise for Poisson phase retrieval. Additionally, we propose an incremental variant of WF, which significantly improves computational efficiency and guarantees convergence to the true signal with high probability under suitable conditions.
Paper Structure (16 sections, 9 theorems, 79 equations, 4 figures)

This paper contains 16 sections, 9 theorems, 79 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem setup and related work
  3. Goals and Motivations
  4. Notations
  5. Organizations
  6. Wirtinger Flow for Poisson phase retrieval
  7. Wirtinger Flow method for Poisson phase retrieval
  8. Convergence result
  9. Incremental Wirtinger Flow for Poisson phase retrieval
  10. Numerical experiments
  11. Step size selection
  12. The influence of ${\mathbf b}$.
  13. Compare with Gaussian model.
  14. Conclusion
  15. Local smoothness condition and local curvature condition
  16. ...and 1 more sections

Key Result

Theorem 2.1

Suppose $y_j=|{\mathbf a}_j^*{\mathbf x}|^2+b_j ,\,j=1,2,\ldots, m$ with $\alpha_1|\langle {\mathbf a}_j, {\mathbf x}\rangle|^2 \leq b_j\leq \alpha_2|\langle {\mathbf a}_j, {\mathbf x}\rangle|^2$. Here $\alpha_1$ and $\alpha_2$ are positive constants. Starting from an initial point ${\mathbf z}_0\in with step size $\mu<2l_{cur}/u^2_{smo}$. Here $l_{cur}$ and $u_{smo}$ are positive constants define

Figures (4)

  • Figure 1: Success rate: The success rate is calculated over 100 trials for varying measurement numbers, with $n = 100$ and $m/n\in[3:0.2:5]$, under noise levels (a) $\eta = 10^{-3}$ and (b) $\eta = 0.1$.
  • Figure 2: Convergence: NRMSE per iteration, with $n = 100$, $m=5n$ under noise levels (a) $\eta = 10^{-3}$ and (b) $\eta = 0.1$.
  • Figure 3: Influence of $b$: record the NRMSE of each iteration step with different $\alpha_1$ with (a) $\eta = 10^{-3}$ and (b) $\eta = 0.1$. Here $n = 100$, $m=5n$, $\alpha_2 = 1/\alpha_1$.
  • Figure 4: Comparison of WF-Gaussian and WF-Poisson: NRMSE is recorded for different measurements under two noise types. (a) and (b) show results under Poisson noise, while (c) and (d) show results under Gaussian noise.

Theorems & Definitions (18)

  • Remark 2.1
  • Theorem 2.1: Exact recovery
  • proof
  • Remark 2.2
  • Theorem 2.2: Stability
  • proof
  • Theorem 3.1
  • proof
  • Lemma A.1: Smoothness Condition
  • proof
  • ...and 8 more